An experienced pilgrim who has all the answers concerning transport in Madrid has posted this information and I am shamelessly stealing it because it is of great value to the pilgrim. Thank you Whariwharangi!
"Shared wisdom ... there is no way you will make it from flight arrival to train departure in the time specified.
Si yo fuera tu ... plan to take a later train."
Getting around in Madrid
The airport in Madrid is called Barajas (MAD). There are 4 terminals. T1, T2, and T3 are located in one building. T4 and T4S are connected by an Automatic People Mover train. There is a shuttle service that connects all terminals.
Madrid Airport (MAD)
Connections to Madrid include Metro, EMT bus, and light rail transit Cercanias.
The Metro has stations at T1-T2-T3 and at T4. The trains run every 6 - 12 minutes. Take Metro line 8 from airport T1-T2-T3 or T4 to Nuevos Ministerios; line 10 to Tribunal; and, line 1 to Renfe Atocha.
Cercanias has a station at T4 and trains run every 30 minutes. Take the Cercanias train from T4 direct to Puerta de Atocha.
EMT Madrid operates a bright yellow express bus every 13 -20 minutes from the airport T1-T2-T3 and airport T4 to Puerta de Atocha with two stops enroute.
Madrid Metro Map
Madrid Cercanias (operated by Renfe). Cercanias trains go direct to Chamartin station and Puerta de Atocha station.
EMT Madrid Airport Express - Airport to Puerta de Atocha
Puerta de Atocha Renfe Station Madrid